Observation of fractional spin textures in a Heusler material

Recently a zoology of non-collinear chiral spin textures has been discovered, most of which, such as skyrmions and antiskyrmions, have integer topological charges. Here we report the experimental real-space observation of the formation and stability of fractional antiskyrmions and fractional elliptical skyrmions in a Heusler material. These fractional objects appear, over a wide range of temperature and magnetic field, at the edges of a sample, whose interior is occupied by an array of nano-objects with integer topological charges, in agreement with our simulations. We explore the evolution of these objects in the presence of magnetic fields and show their interconversion to objects with integer topological charges. This means the topological charge can be varied continuously. These fractional spin textures are not just another type of skyrmion, but are essentially a new state of matter that emerges and lives only at the boundary of a magnetic system. The coexistence of both integer and fractionally charged spin textures in the same material makes the Heusler family of compounds unique for the manipulation of the real-space topology of spin textures and thus an exciting platform for spintronic and magnonic applications.


Magnetic textures and competing interactions
Previous studies have pointed out that the dipolar interaction plays an important role in Heusler compounds such as Mn1.4Pt0.9Pd0.1Sn 1,2 . This is particularly prominent at low temperatures, where elliptically deformed Bloch skyrmions are stabilized despite the anisotropic Dzyaloshinskii-Moriya interaction that favors antiskyrmions 1 . In our studies, at higher temperatures -room temperature and above -we do not observe these elliptically deformed Bloch skyrmions but rather only antiskyrmions as the concomitant decrease in the total magnetization of the sample suppresses the dipolar interactions. Therefore, this Heusler material provides a unique opportunity to study the competition between the Dzyaloshinskii-Moriya interaction, dipolar interactions, as well as the easy-axis anisotropy, at finite temperatures.
We have searched for parameters within our atomistic spin Hamiltonian model (see Methods) that can qualitatively reproduce the experimental results of the main text. Our system consists of 50 × 50 × 5 spins with free boundary conditions, "# = 0.05, and + -/ = 0.05. To efficiently perform Monte Carlo simulations, we employ the stochastic cut-off (SCO) method adapted for long-range dipolar interactions, which drastically reduces the computational cost for three-dimensional systems [3][4][5] . Our bond-switching algorithm for implementing the SCO method is based on Ref. 5. Starting from a thermal configuration, the temperature is lowered from / = 20 to / = 0.7 in 30 steps, each one consisting of 20,000 Monte Carlo steps.
The magnetic configuration for each parameter set { / , / } is obtained by taking the thermal average over 1000 samplings. Fig. S2a-d show the magnetic textures obtained with increasing easy-axis anisotropy for / = 0.07. When the easy-axis anisotropy is sufficiently small, the in-plane vortex phase is the most stable configuration and carries no topological charge. We conclude that the easy-axis anisotropy of our Mn1.4Pt0.9Pd0.1Sn samples must be sufficiently large since no in-plane vortices were revealed by LTEM imaging. As the easy-axis anisotropy is increased, various topological objects are stabilized. Crucially, elliptical Bloch skyrmions are first stabilized at / = 0.35, which have a swirling spin texture like the vortex phase as shown in Fig. S2b. Further increasing the easy-axis anisotropy favors the formation of antiskyrmions. Non-topological bubbles are obtained as a metastable state at the boundary between antiskyrmions and elliptical Bloch skyrmions. Therefore, our study suggests that the easy-axis anisotropy also plays an important role in stabilizing elliptical Bloch skyrmions in Heusler materials.
In Fig. S2e we summarize our results on the stability of different topological spin structures as a function of the Dzyaloshinskii-Moriya interaction and easy-axis anisotropy , obtained by single-shot Monte Carlo simulations. We conclude that the stability of elliptical Bloch skyrmions and antiskyrmions is determined by the competition among the dipolar interaction, Dzyaloshinskii-Moriya interaction, and easy-axis anisotropy. In particular, for / < 0.08, Bloch skyrmions could coexist with antiskyrmions depending on the strength of the easy-axis anisotropy. Since the demagnetization field becomes larger at low temperatures due to an increased saturation magnetization, elliptical Bloch skyrmions become more favourable than antiskyrmions at low temperatures in Heusler materials.

Stability of fractional (anti)skyrmions
As discussed in the main text, the ferromagnetic phase in chiral magnets becomes unstable against nucleation of helical domains at the edges of the sample below the critical magnetic field. To confirm the stability of edge magnetic textures in systems with 9: symmetry and dipolar interaction, we studied the energy landscape of fractional skyrmions as a function of the applied magnetic field fixing the other magnetic parameters at { / , / , "# } = {0.05, 0.45, 0.05} . Using the same protocol of efficient Monte Carlo simulations as in the previous section, we prepared the spin configuration of an antiskyrmion, an elliptical Bloch skyrmion, a fractional antiskyrmion, and a fractional Bloch skyrmion in a 50 × 50 × 5 spin system with free boundary conditions at / = 0.7. In order to evaluate the energy of each metastable configuration, the temperature was abruptly dropped to zero and then each object was relaxed employing the Landau-Lifshitz-Gilbert equation for 2,000 time steps.
The energy was computed as a function of magnetic field by slowly changing the magnetic field over 2,000 time steps and undergoing further relaxation for another 2,000 time steps. We should note that the stability between antiskyrmions and elliptical Bloch skyrmions depends on the relative strength of / and / in comparison to the value of "# . In fact, we find that antiskyrmions become energetically more favourable than elliptical Bloch skyrmions at / = 0.07 and / = 0.45 for all magnetic fields.

Dynamics of conversion, annihilation, and merging
To further investigate the conversion and annihilation mechanisms as the magnetic field is varied, we performed atomistic spin dynamics simulations. An initial spin configuration was

Connection between real-and reciprocal-space topologies in (anti)skyrmion crystals
Recently, it has been shown that (anti)skyrmion crystals support a higher-order topological phase characterized by a nontrivial magnonic quadrupole moment 9 , protected by crystalline symmetries and whose hallmark signatures are robust magnonic corner states. Using nested Wilson loops adapted to magnonic systems, this novel topological phase was found to exist for a broad range of magnetic fields. However, in order to realize these magnonic corner states, it is essential to preserve the protecting symmetries at the sample boundaries. This is particularly important in magnetic systems, as the magnetization profile near the edges of the sample could be easily distorted from that of the magnetic unit cell in the bulk of a sample.
A crucial finding in Ref. 9 is that the classical spin ground state in a finite sample exhibits a reconstruction with fundamental consequences for the magnonic topological excitations. As discussed in the main text, the twisted spin configuration along the sample edges facilitates the formation of fractional (anti)skyrmions below the critical magnetic field. The mutual repulsion between fractional (anti)skyrmions and bulk (anti)skyrmions results in their self-assembly that restores the symmetries protecting the bulk topology in reciprocal space. As an important consequence, the topological behaviour of the magnon states in the bulk gives rise to special magnon edge states at the sample boundaries. This is a manifestation of the celebrated bulk-boundary correspondence. Without fractional (anti)skyrmions there would be no such edge states and the bulk-boundary correspondence would not hold. In particular, for the system under consideration in-gap magnon states with large probability amplitude are predicted to emerge at the boundary, more precisely, at each corner of the sample. In other words, fractional

Formation of fractional antiskyrmions at low temperature
In our Mn1.4Pt0.9Pd0.1Sn sample, we find that antiskyrmions are stable at room temperature and above, whereas elliptical Bloch skyrmions are found at low temperatures 1 . However, one is able to observe antiskyrmions at low temperatures using the following field cooling procedure, as schematically shown in Fig. S5a. On stabilizing a square antiskyrmion lattice at room temperature in the presence of a constant perpendicular magnetic field, we reduce the specimen's temperature to 100 K.  S5d). The size of these fractional nano-objects reduces as the field crosses zero and increases in strength in the negative direction ( Fig. S5e and S5f). The fractional antiskyrmions become triangularly-shaped in the field region from -72 mT to -224 mT (Fig. S5g and S5j). Fig. S1. Lamella preparation using focused ion beam milling. Lamellae specimens used for LTEM imaging are formed from bulk polycrystalline samples of Mn1.4Pt0.9Pd0.1Sn using focused ion milling. An individual grain, that is single crystalline, is identified whose crystal orientation is such that the [001] axis is perpendicular to the lamella that will be formed. EBSD   where the topological charge density ( ) for a discrete lattice is defined in Ref. 8. In some cases, we obtain = 2 for two antiskyrmions and = 1/2 for edge fractional antiskyrmions.

Supplementary figures
All the results are obtained in a 50 × 50 × 5 spin system using Monte Carlo simulated annealing with "# = 0.05, + -/ = 0.05, and / = 0.7.    This region is square shaped. Secondly, the same yellow rectangular region identified in Fig.   S4, that is representative of the interface between Mn1.4Pt0.9Pd0.1Sn and PtCx. (a5-a9, b5-b9, c5-c9, d5-d10) The interior and interface regions are shown at various magnetic fields. The number of fractional nano-objects formed at the interface depends on the density of the nano-objects in the lamella. The higher the density of these objects, the larger is the number of fractional objects at the interface. The dimension of the square shaped interior region is 2500 nm × 2500 nm and the interface region has dimensions of 1300 nm × 6000 nm.